I have to give $5$ examples of $D_5$ acting on a set. So far, I have $D_5$ acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection in the $x$-axis, rotating the edges of a pentagon one to the right and reflecting the edges in the $x$-axis but I’m struggling to come up with a fifth example. Could someone help me out with a final example and/or tell me if I’ve made any mistakes with the examples I’ve given?
If I understood your question correctly, you listed 3 examples of $D_5$ acting on vertices of a pentagon. Here's how to get more.
Let $\rho$ be an element of order $5$ in $D_5$ and let $\sigma$ be an element of order $2$ in $D_5$. One action of $D_5$ on vertices of a pentagon would correspond to $\rho$ rotating the pentagon by $72^\circ$ and $\sigma$ reflecting it.
There are also actions corresponding to where $\rho$ rotates the pentagon by any other multiple of $72^\circ$ (including $0^\circ$).
There are also actions where $\rho$ rotates the pentagon but $\sigma$ does nothing to the pentagon.
You could also consider actions of $D_5$ on itself (e.g., multiplication or conjugation).
The most important action that is usually forgotten is the identity operation. You keep all the vertices in place. It's an important one, because element “one” is literally in the definition of a group.
I think you might be good with this.